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This essential trains for: SAT-II, AMC-8, AMC-10, Math Kangaroo 9-10.

The Law of Sines

The area of a scalene triangle ABC can be computed using two sides and the angle between them:

Consider the triangle with side lengths a, b, c and angle B formed by the sides a and c. If we could compute the height corresponding to side c from this information, we could use the previous formula to find the area.

We can achieve this by using the definition of the sine ratio:

Solving for H:

we can substitute this expression for H in:

Repeating the same procedure with other pairs of sides:

Multiplying all the ratios by 2:

Dividing by the product of all side lengths:

where from we get:

an identity which is also known as the law of sines.

The law of sines is useful when the data provided in the problem consists of a side length and measures of the two angles adjacent to it.

The Law of Cosines

Consider a scalene triangle in which we know the lengths of two of the sides and the measure of the angle adjacent to them.

If we are interested in computing the length of the third side, we can use an additional construction. In the triangle ABC draw the altitude AD:

The goal is to find the length of AC (b). We cannot apply the Pythagorean theorem in a scalene triangle, but we can apply it twice in the two right angle triangles ABD and ADC.

Denote:

Use the Pythagorean theorem in the triangle ABD:

Use the definition of the cosine ratio to replace BD:

Use the Pythagorean theorem in the triangle ADC:

Use the definition of the cosine ratio to replace CD:

Now set the expressions for h2 to be equal:

and do the algebra:

Re-arranging the terms we get:

which is known as the law of cosines.

By permutations, the following are also true: